We present a unified analysis for a class of long-step primal–dual path-following algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central path HP(XS) ≡ [PXSP–1 + (PXSP–1)T]/2 = μI, introduced by Zhang. At an iterate (X,S), we choose a scaling matrix P from the class of nonsingular matrices P such that PXSP–1 is symmetric. This class of matrices includes the three well-known choices, namely: P = S1/2 and P = X–1/2 proposed by Monteiro, and the matrix P corresponding to the Nesterov–Todd direction. We show that within the class of algorithms studied in this paper, the one based on the Nesterov–Todd direction has the lowest possible iteration-complexity bound that can provably be derived from our analysis. More specifically, its iteration-complexity bound is of the same order as that of the corresponding long-step primal–dual path-following algorithm for linear programming introduced by Kojima, Mizuno and Yoshise.
